Question (idea): Is there a notion of tensor product of chain complexes in a $\mathcal{V}$-enriched monoidal category $\mathcal{C}$, for $\mathcal{V}$ a linear symmetric monoidal category?
Let me detail the question. We fix $\mathcal{V}$ a linear (over some commutative ring) symmetric monoidal category. Because $\mathcal{V}$ is monoidal, there is a notion of (small) $\mathcal{V}$-enriched categories, forming a category $\mathcal{V}\text{-}\mathcal{C}at$. Because $\mathcal{V}$ is symmetric, $\mathcal{V}\text{-}\mathcal{C}at$ admits a tensor product, and we can define the notion of a $\mathcal{V}$-enriched monoidal category as a monoid internal to $\mathcal{V}\text{-}\mathcal{C}at$; see the nLab.
Let $(\mathcal{C},\otimes)$ be a $\mathcal{V}$-enriched monoidal category. Because $\mathcal{V}$ is assumed to be linear, the notions of chain complexes, chain morphisms and homotopies carry verbatim to $\mathcal{V}$-enriched categories, and in particular the category $\mathrm{Kom}(\mathcal{C})$ of chain complexes and chain morphisms make sense.
Question (precise): Is there a monoidal structure (enriched or not) on $\mathrm{Kom}(\mathcal{C})$, induced by the $\mathcal{V}$-enriched monoidal structure on $\mathcal{C}$?
This should extend the usual product of complexes on linear categories, ie when $\mathcal{V}=(\mathrm{vec},\otimes,\beta)$ is the category of finite vector spaces equipped with the standard monoidal structure and braiding. Possibly, one would need to restrict to a subcategory $\mathcal{D}\subset\mathrm{Kom}(\mathcal{C})$. Ideally, this tensor product should be compatible with homotopies, in the sense that if $A_\bullet\simeq B_\bullet$ and $C_\bullet\simeq D_\bullet$, then $A_\bullet\otimes C_\bullet\simeq B_\bullet\otimes D_\bullet$ (where $\simeq$ means "being homotopic").
Additional remarks:
- It is probably sufficient to replace, in the axioms of $\mathcal{V}$, symmetric by braided, and linear by pointed.
- For my purpose, it is fine to only consider the strict version of a $\mathcal{V}$-enriched monoidal category.
